Let $(X,d)$ be a metric space and $(K_X , h_d)$ be the associated metric space of nonempty compact subsets of $X$ with the Hausdorff metric. It is well known that $K_X$ inherits certain topological (and analytic) properties from $X$. For example, if $X$ is compact, then so is $K_X$; and if $X$ is complete, then so is $K_X$.

Is there a reference which further explores the properties that $K_X$ inherits from $X$? In particular, if $X$ is locally compact then is $K_X$ also?

The associated metric space of compact subsets of đť‘‹ with the Hausdorff metric": you mean ofnonemptycompact subsets. Otherwise I don't see how you measure the distance to the empty set. For $X$ is locally compact, you can indeed make the set of compact subsets (includingthe empty set) topological in a natural way, and $\emptyset$ is isolated in this space iff $X$ is compact; in all cases the space of nonempty compact subsets is locally compact. $\endgroup$3more comments